The generators \(P_{m,n}\)  transform under global conformal symmetry like modes of a primary operator with weight \((h, {\bar{h}}) = \left(\frac{3}{2}, \frac{3}{2}\right)\)  [1]}, [2]}
\( \begin{split}\left[L_k, P_{m,n}\right] &= \left(\frac{1}{2}k-m\right)P_{m+k,n}, \quad \quad \quad \left[\bar{L}_k, P_{m,n}\right] = \left(\frac{1}{2}k-n\right)P_{m,n+k},\end{split}\) 
